Obviously something has shifted. Likely it's the driveway rather than your garage, and if it's shifted, it will likely continue to shift farther away. So, a permanent fix likely will work itself out farther, will look bad, and will assuredly crack. The expensive solution would be to figure out why the driveway is shifting, fix that, and then re-pour the section of concrete next to your garage to create a tight seal.
But, if it were me, I'd fill that gap with some sort of backer rod and then cover it with flexible concrete seam sealer - all of which likely can be obtained from a local big-box store. And that popped out section, I'd just fill it with seam sealer and call it "good enough". The driveway will continue to shift and you'll need to refresh/replace the seam sealant as needed, but that will be a lot cheaper than replacing the driveway.
If you can't get backer rod that wide, maybe using some foam pool noodles might fill that gap nicely.